Removing brackets 1
In order to simplify mathematical expressions it is frequently
necessary to remove brackets. This means to rewrite an expression
which includes bracketed terms in an equivalent form, but without
any brackets. This operation must be carried out according to
certain rules which are described in this leaflet.
1. The associativity and commutativity of multiplication
Multiplication is said to be a commutative operation. This
means, for example, that 4Ã—5 has the same value as 5Ã—4.
Eitherway the result is 20. In symbols, xy is the same as yx, and
so we can interchange the order as we wish. Multiplication is
also an associative operation. This means that when we want to
multiply three numbers together such as 4Ã—3Ã—5 it doesn matter
whether we evaluate 4Ã—3 first and then multiply by 5, or
evaluate 3Ã—5 first and then multiply by 4. That is (4Ã—3)Ã—5 is
the same as 4Ã—(3Ã—5) where we have used brackets to indicate
which terms are multiplied first. Eitherway, the result is the
same, 60. In symbols,we have (x Ã— y) Ã— z is the same as x Ã— (y
Ã— z) and since the result is the same eitherway, the brackets
make no difference at all and we can write simply x Ã— y Ã— z or
simply xyz. When mixing numbers and symbols we usually write the
numbers first. So
Example
Remove the brackets from
a) 4(2x)
b) a(5b)
Solution
a) 4(2x) means 4Ã—(2 Ã— x). Because of associativity of
multiplication the brackets are unnecessary and we can write 4Ã—2
Ã— x
b) a(5b) means a Ã—(5b). Because of commutativity this is the
same as (5b) Ã— a that is (5 Ã— b) Ã— a. Because of associativity
the brackets are unnecessary and we write simply 5 Ã— b Ã— a
which equals 5ba. Note that this is also equal to 5ab because of
commutativity.
Exercises
1. Simplify
Answers
2. Expressions of the form a( b + c ) and a( b  c )
Study the expression 4Ã—(2 + 3).
By working out the bracketed term first we obtain 4Ã—5 which
equals 20. Note that this is the same as multiplying both the 2
and 3 separately by 4, and then adding the results. That is 4Ã—(2
+ 3) = 4Ã—2 + 4Ã—3 = 8 + 12 = 20.
Note the way in which the "4" multiplies both the
bracketed numbers, "2" and "3". We say that
the "4" distributes itself over both the added terms in
the brackets  multiplication is distributive over addition.
Now study the expression 6Ã—(8  3).
By working out the bracketed term first we obtain 6Ã—5 which
equals 30. Note that this is the same as multiplying both the 8
and the 3 by 6 before carrying out the subtraction:
6Ã—(8  3) = 6Ã—8 6Ã—3 = 48  18 = 30.
Note the way in which the "6" multiplies both the
bracketed numbers. We say that the "6" distributes
itself over both the terms in the brackets  multiplication is
distributive over subtraction. Exactly the same property holds
when we deal with symbols.
a (b + c) = ab + ac
a (b  c) = ab  bc
Example
Exercises
Remove the brackets from each of the following expressions
simplifying your answers where appropriate.
Answers
